Heilbronn triangle problem

In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points within a region in the plane, in order to avoid triangles of small area. It is named after Hans Heilbronn, who conjectured prior to 1950 that this smallest triangle area is necessarily at most inversely proportional to the square of the number of points. Heilbronn's conjecture was proven false, but the asymptotic growth rate of the minimum triangle area remains unknown.

The problem may be defined in terms of any compact set D in the plane with nonzero area such as the unit square or the unit disk. If S is a set of n points of D, then every three points of S determine a triangle (possibly a degenerate one, with zero area). Let Δ(S) denote the minimum of the areas of these triangles, and let Δ(n) (for an integer n ≥ 3) denote the supremum of the values of Δ(S).

The question posed by Heilbronn was to give an expression, or matching asymptotic upper and lower bounds, for Δ(n). That is, the goal is to find a function f, described by a closed-form expression, and constants c₁ and c₂, such that for all n,

In terms of big O notation, the left inequality may be written as Δ(n) = Ω(f(n)), the right inequality may be written as Δ(n) = O(f(n)), and both of them together may be written as Δ(n) = Θ(f(n)). The shape and area of D may affect the exact values of Δ(n), but only by a constant factor, so they are unimportant for its asymptotic growth rate.

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